Mathematics has been around for thousands of years, and this has given it plenty of opportunities to become very complicated. For example, seemingly disparate fields of mathematics have often, over time, become connected in surprising ways. In his book Love and Math, Edward Frenkel describes this process via an analogy with continents and the bridges that are built between them. This is very apt, since building a bridge between (say) North America and Europe would be very difficult, and much of the mathematics he writes about is also very difficult.

The book is partially Frenkel’s autobiography: it describes how he got involved with mathematics, the problems he faced as a Jew in the old Soviet Union and his love of the subject. In order to convey this history, he also explains a great deal of mathematics, and particularly that associated with the Langlands programme, which aims to unify the different branches of the field. This mathematics is sophisticated (I will briefly present some later), yet Frenkel manages to give even relatively inexperienced readers a sense of it. In particular, he conveys that this material is interesting and important. While I was reading the book, I was also working on a fun but frankly non-important problem in mathematics. The contrast was striking!

The book – which is written in the first person – begins by describing some elementary particle physics and group theory and explaining how they relate to each other. After this comes the author’s own tale. Frenkel was born in 1968 in what was then the Soviet Union, and he initially wanted to study physics. His mentor, Evgeny Evgenievich Petrov, converted him to mathematics, the subject in which he earned his PhD, but since his work is often related to physics it is not clear he really did switch. At such high levels it is hard (and not productive) to distinguish between the two.

As a young man, Frenkel was a brilliant mathematician, and if the Soviet Union had not practised a form of institutionalized antisemitism, he would have passed his exams and got into Moscow State University. But under the circumstances, such success was impossible no matter how well he performed. The examiners kept making the questions harder and harder, both in terms of their intellectual merit (he was still able to solve questions of this type) and in terms of stupid pedantics. As an example, Frenkel defined a circle as “the set of points equidistant from a point”, but his examiners deemed this answer wrong since it should be “the set of all points equidistant from a point”. Another way for Jews to be barred from school was to give them harder problems to solve. Often such problems had an easy solution that was hard to find, thus giving the appearance of fairness. (For more on this see “Jewish problems” by T Khovanova and A Radul, arXiv:1110.1556.)

Frenkel’s experience was far from unique, and in his book he describes the ways in which many Jewish mathematicians, physicists and other scientists dealt with this system. Some of them met quasi-secretly and still managed to get much done, and it is tempting to wonder whether oppression got their creative juices flowing. However, this is a fallacy. We only read about those who managed to do well, and I am sure that many brilliant students were blocked from making contributions. Their biographies are not written.

At the heart of the book is Frenkel’s description of the Langlands programme, which aims to build those “bridges” between different mathematical “continents”. Here is an example. Let p be a natural (or “counting”) number. If we restrict ourselves to the set of numbers {0,1,2…,p–1} then we can still add, subtract, and multiply if we “wrap around” back to the beginning. For example, if p = 13, then 12 + 4 = 3, which we denote as 12 + 4 ≡ 3 (mod 13). Also note that 6 + 7 ≡ 0 (mod 13), so we think of 7 as being –6 (mod 13). If p is a prime number, then we can also divide.

Now, let f(x,y) be a cubic polynomial in two variables with integer coefficients, such as f(x,y) = y³ + y – x³ – 2x². If p is prime, we can ask how many pairs (a,b) with (a,b) included in the set {0,1,2…p–1} exist such that f(a,b) ≡ 0 (mod p). We denote this number as n_p. We can then form an infinite polynomial p(x) where n_p is the coefficient of x^p. This infinite polynomial is then associated to a group of symmetries in the complex plane G called a modular form. The correspondence between f(x,y) and its modular form is one-to-one and it preserves some properties; that is, every cubic equation maps to a modular form and every modular form maps to a cubic equation. This is an important connection.

What the Langlands programme does is essentially to take the notion of equation and generalize it, and also to take the notion of modular form and generalize that. The programme then makes conjectures about how these very general objects are related. Frenkel illustrates this connection-building process with several nice examples until, on page 222, he has a chart that connects number theory, Riemann surfaces (geometry) and quantum physics. Quantum physics? How did that get in there? Through gauge theory – a complicated notion that Frenkel (wisely) does not try to explain. However, having read the book, I now want to find out what it is.

Frenkel claims that frequently, a branch of mathematics that was thought of as a pure abstraction ends up being applied to practical problems. I am often sceptical of such claims, since the (perhaps forgotten) origin of many mathematics problems is, in turn, some real-world application. However, the examples given here seem legitimate; to my eye, at least, number theory really does lack any apparent connection to the practicalities of quantum physics, yet the links are there. One is left with the impression that Frenkel and the other scholars who appear in the book (including Ed Witten, the only physicist to win a Field’s Medal) are seriously brilliant people who are doing seriously brilliant work.

You do not need to know much mathematics to read this book, but you do need to like it. Depending on your level, you will get lost at some point (for me, it was the definition of a “sheaf”). However, this is not a book to read to learn maths. It’s a book to read to be inspired to learn maths.